Tran T. Applications of the Kloosterman sums in cryptography and coding

The thesis deals with applications of the Kloosterman sums and its generalizations over the ring of rational integers and Gaussian integers in cryptography and coding. There are obtained the new results on properties these sum. For multiplicative function of special type whict weighted by the Kloosterman sums it is constructed the representations of summatory functions in terms of zeta-similarly functions. For the incomplete Kloosterman sums over the ring of integers there are obtained estimates of adequate belinear forms. For construct of the sequences of pseudorandom numbers there are embedded new congruential linear-inversive generators. The discrepancy of pseudorandom numbers that produced by linear-inversive generators estimate there is estimated with the help of the estimates of the Kloosterman sums. It is proved that such sequences of pseudorandom numbers passe the serial test on equidistribution and unpredictability. And analogue of power congruential generator of the sequence of complex numbers from unit circl there is constructed on the norm subgroup of the residue ring modulo , . It is investigated a distribution of values of the weight function of the Kloosterman code over the residue ring modulo , of the Gaussian numbers.